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3.7.1 Student's t Distribution

3.7.1 Student's t Distribution

Key Concepts

  • t-Distribution: A family of distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
  • Degrees of Freedom: Related to the sample size; affects the shape of the distribution.

Detailed Explanation

Formation of t-Distribution

  • A t-distribution arises when a standard normal variable Z is divided by the square root of a Chi-square variable X scaled by its degrees of freedom k:

    • T = Z / sqrt(X/k)

  • Degrees of Freedom: The parameter k in the Chi-square variable determines the degrees of freedom for the t-distribution.

Properties of t-Distribution

  • Mean and Variance: The mean of the t-distribution is 0 (for k > 1), and its variance is k / (k-2) (for k > 2), making it wider and flatter than the normal distribution.
  • Shape: Similar to the normal distribution but with heavier tails, providing greater flexibility for analyzing data with outliers or small samples.
  • Convergence to Normality: As k increases, the t-distribution converges to the normal distribution.

Applications

  • Hypothesis Testing: Particularly useful for estimating means from small sample sizes, especially in student t-tests.
  • Confidence Intervals: Used to calculate the confidence intervals for means when the sample size is small and the population variance is unknown.

Practical Application

Example Calculation:

  • If you have a sample of size 10 from a normal distribution and need to test the hypothesis about the mean, you would use a t-distribution with 9 degrees of freedom to determine the test statistics and p-values.
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